Optical properties of any material can be described by the complex index of refraction, N=n-ik, or the complex dielectric function, .epsilon.=.epsilon..sub.1 -i.epsilon..sub.2. .epsilon. is related to N by .epsilon.=N.sup.2, so that .epsilon..sub.1 and .epsilon..sub.2 can be determined from a knowledge of n and k: .epsilon..sub.1 =n.sup.2 -k.sup.2 and .epsilon..sub.2 =2nk. The real and imaginary parts of the complex index of refraction, n and k, are termed refractive index and extinction coefficient respectively. In addition to k and .epsilon..sub.2, which relate to absorption of light, the absorption coefficient, .alpha.=2.omega.k/c, is used to describe absorption where .omega. represents photon frequency and c, the speed of light and where n and k, .epsilon..sub.1 and .epsilon..sub.2, and .alpha. are referred to as optical constants of the material. Values of optical constants depend on photon energy, E=.omega.; that is, N=N(E)=n(E)-ik(E), .epsilon.=.epsilon.(E)=.epsilon..sub.1 (E)-i.epsilon..sub.2 (E), and .alpha.=.alpha.(E). These functions are called optical dispersion relations.
Electrical properties of a material can be found from its optical properties. For example, a.c. conductivity, .sigma., is directly proportional to .epsilon..sub.2, through .sigma.=E/h .epsilon..sub.2 =2nkE/h where h is Planck's constant. Furthermore, d.c. conductivity, .sigma..sub.0, is proportional to a time parameter, .tau., representing how long (on the average) an electron remains in the conduction band. This time parameter .tau. can be taken as the lifetime of the excited state to which the electron transfers due to optical (photon) absorption. Thus, .sigma..sub.0 =(N.sub.f e.sup.2 .tau.)/m.sub.e where N.sub.f is the number of free electrons per unit volume, e, the electron charge, and m.sub.e, the electron mass.
In what follows, explicit consideration will be given to determination of n and k with the understanding that .epsilon..sub.1 and .epsilon..sub.2 can be obtained from this determination.
Previous formulations of optical dispersion relations vary for different materials, are complicated, and use different equations for different energy ranges. These previous formulations determine optical constants for semiconductors and dielectrics in closed analytical form only for a narrow range of photon energies just above the energy band gap, E.sub.g, that is, for the absorption edge. In this range it is postulated that .alpha. as well as k and .epsilon..sub.2 (since n is assumed constant for E=E.sub.g) can be broken into two parts. For .alpha.&lt;.alpha..sub.0 (where .alpha..sub.0 represents some experimentally determined cut-off value), an empirical exponential dependence on E, referred to as Urbach's tail, is assumed. For .alpha.&gt;.alpha..sub.0, a power law dependence on E is derived, based on a one electron model with infinite lifetime for the excited electron state. The value of the exponent depends on the type of transition: direct, forbidden, or indirect.
Beyond the absorption edge, optical constants are not determined in closed analytical form. Analysis of critical points in the Brillouin zone plays a major role in their determination. For example, an edge in the .epsilon..sub.2 vs E spectrum is attributed to a transition corresponding to .omega.=E.sub.cv (k.sub.crit). At a critical point, k=k.sub.crit, the function E.sub.cv (k)=E.sub.c (k)-E.sub.v (k) is an extremum. (k represents electron wave vector and indices c and v represent conduction and valence bands respectively.) A peak in .epsilon..sub.2 vs E is attributed to a transition corresponding to an accidental pairing of two critical points occurring near the same energy (sometimes enhanced by exciton interactions).
The principle of causality leads to a fundamental relation between the real, n(E), and imaginary, k(E), parts of N(E), the Kramers-Kronig relation. Thus, theoretically, when k as a function of E is known, n(E) can be determined. However, except for harmonic oscillator fits to the measured data, formulations of n(E) do not relate it to k(E). Instead, n(E) is determined empirically by fitting data to various model equations. Commonly used are Sellmeier type equations or equations involving a sum of Sellmeier terms, all valid for a limited range of energies. The particular equation applied to a given material is determined by the resulting fit it gives to measured data.
When an oscillator fit is used, n can be related to k through the Kramers-Kronig relation. However, k cannot be correctly described at the absorption edge since the energy band gap is not incorporated into an oscillator model. In addition, many fitting parameters are required. For example, 22 parameters are used to describe n and k for crystalline Si in the 0 to 10 eV range.
For metals, two major interactions are commonly used to account for their optical properties. In the far and near infrared regime, a classical free electron-electromagnetic interaction is used, that is, the Drude free electron model. In the fundamental optical regime, quantum mechanical absorption of photons accounts for the interaction.
The low energy classical free electron model seldom agrees precisely with experiment. It is often found that the measured conductivity is less than the predicted classical value and that the current density and electric field are not in phase so that Ohm's law is not obeyed in the optical interactions. Also, some structure can be discerned in the optical constants. Furthermore, the Hagens-Ruben relation is violated in many instances with n&gt;k for very low energies. Several factors are held responsible for these discrepancies. One factor involves the anomalous skin effect whereby electrons near the surface have frequent collisions and therefore a shorter mean free path and smaller effective relaxation time that electrons deeper in the metal. Since the electromagnetic waves penetrate only a very short distance in the metal, they only interact with surface electrons. A second reason for disagreement is ascribed to bound electron transitions which influence the optical properties and yet do not provide any electrical conductivity. A third reason is attributed to multiple frequency dependent relaxation times for free electrons.
In the fundamental optical regime (where structure in optical constants is apparent for most materials), quantum mechanical absorption of photons accounts for the interaction. Two different quantum approximations can be found. The one-electron model approximates each electron as an independent particle where the random phase approximation takes into account the electron-electron interaction. An infinite lifetime of the excited state to which an electron transfers via photon absorption is mostly assumed in these approximations. Some treatments, however, do incorporate an interband relaxation time parameter in the expression for .epsilon.. Formulations stemming from these approximations provide structural details for .epsilon..sub.2 (E) such as thresholds, edges, peaks and saddle points through analysis of critical points of the Brillouin zone. These treatments, however, do not provide an analytical expression for .epsilon..sub.2 (E) (or k(E)) applicable throughout the optical regime and thus cannot give .epsilon..sub.1 (E) (or n(E)) as stipulated by the Kramers-Kronig relations. .epsilon..sub.1 (E) (as well as n(E)) are treated from a phenomenological point of view. Moreover, they do not resolve the issue of direct transitions (in which electron momentum is conserved), versus indirect transitions (where phonons are absorbed or emitted in order to conserve momentum) in metals.
Currently, there are several methods to determine the optical constants of a medium (bulk and thin film). See, e.g., "Handbook of Optical Constants of Solids", edited by E. D. Palik (Academic Press, N.Y., 1985), and O. S. Heavens, in "Physics of Thin Films", Vol. 2, edited by G. Hass and R. E. Thun (Academic Press, N.Y., 1964), for reviews of various methods. These methods of determining n and k, however, are complicated and at times yield inaccurate results.
For bulk materials, a widely used method of determining n and k is by Kramers-Kronig (K-K) analysis of the reflectance spectrum. Reflectance, R, is defined as the ratio of the intensity of the reflected beam, I.sub.R, to that of the incident beam, I, that is, R=I.sub.R /I. In principle, optical constants of bulk materials as functions of energy can be determined by K-K analysis if R(E) is known for all energies from zero to infinity (0 to .infin.). In practice, however, R is measured only over a limited range of energies and extrapolated beyond the range of measurements. These extrapolations introduce errors in determination of the optical constants. In addition, although there are different schemes to extrapolate R to infinity, they are intrinsically flawed because they all assume R(.infin.)=0. This is because in classical theory of dipersion it is assumed that n(.infin.)=1 and k(.infin.)=0, which implies that R(.infin.)=0; whereas, as described later, it is shown that n(.infin.).noteq.1 and k(.infin.).noteq.1, which implies that R(.infin.).noteq.0.
Another method of determining the optical constants of a medium is by ellipsometry. In this method, n and k are determined by measuring the change in the state of polarization of reflected light. Ellipsometry requires complex instrumentation, and needs certain sophistication in interpretation of the measurements. Moreover, alignment of the polarizing elements is very important and can lead to errors. In addition, ellipsometry is rarely carried out at energies above 6 eV for lack of effective polarizing elements.
For thin films, n and k at each wavelength are currently determined from two independent measurements made at those wavelengths, such as the reflectance R and transmittance T of the same film (RT method), or transmittance of two films of different thicknesses (TT method). (Transmittance is defined as the ratio of transmitted to incident intensity.) In all these cases, the thickness of the film must be determined by other means. A disadvantage of the RT method is that the substrate upon which the film is deposited must be transparent in order to make the transmittance measurement possible. This limits the choice of substrate for deposition of thin films. (Note that properties and growth rates of thin films can be significantly altered depending on the type of substrate. In fact, the dependence of the film properties on the type of substrate is sometimes used for selective deposition of thin films.) An obvious disadvantage of the TT method, that is, using two films of different thicknesses for determination of the optical constant, is that the method assumes no variation of the optical constants with thickness, which is not usually the case.
It is an object of the present invention to provide a simple and accurate method and apparatus for determining the optical constants of materials based on new expressions for n and k. It is another object of the present invention to provide a method and apparatus of determining simultaneously the optical constants and the thicknesses of thin films.
A further object of the present invention is to determine a.c. and d.c. conductivity for materials.